Question

In a solid, oxide ions are arranged in ccp. One sixth of the tetrahedral voids are occupied by the cations X and one third of the octahedral voids are occupied by the cations Y. What is the formula of the compound?

Answer 1

Hint: We need to know about unit cells, their types and how are atoms arranged in the unit cell. Crystalline solid with regular and repeating pattern of constituent particles in a three-dimensional arrangement which is called crystal lattice. In this crystal lattice, the smallest portion which when repeated in different directions produces the lattice is called the unit cell of that crystalline solid.

Complete answer:
Unit cells are divided into two categories – primitive and centered unit cells. When the
constituent particles are present only on the corner positions then it is called a primitive unit cell. When one or more constituent particles are present in other positions in addition to those at corners are called centered unit cells. Centered unit cells are of three types:
1.Body-Centered Unit Cells: Such unit cell contains one constituent particle at its body-Center besides the one present at the corners.
2.Face- Centered Unit Cells: Such unit cells have one constituent particle present at the Center of each face, besides the ones that are at its corners.
3.End Centered unit cells: Such unit cells have one constituent particle at the Center of any two opposite faces besides the ones present at its corners.
There in total seven crystal systems and primitive unit cells are a part of all the seven systems. However, the three centered unit cells are a variation of a few of them. We shall discuss only one of the systems associated with diamonds which is face-Centered cubic unit cells, also known as fcc unit cells. Cubic Close Packed (CCP) or Face Centered Cubic (fcc) is the same lattice known by two different names.
A fcc unit cell contains atoms at all the corners and the centre of all the faces of the cube and each atom located at the face-center is shared between two adjacent cells and only half of each atom belongs to one unit cell.
The 8 corners of the cubic unit cell contribute only $\dfrac{1}{8}$ atom per unit cell.
When the unit cells are closely packed, one layer of unit cells above another creates a tetrahedral void which is occupied by another carbon atom which contributes only $\dfrac{1}{2}$ of its atom to one unit cell. Hence there are $8$ voids created by the closed packing arrangement. In addition, there are $4$ octahedral voids created in this kind of arrangement.
The effective number of oxide ions =4(CCPstructure)
Effective number of cation X =$\dfrac{1}{6} \times $​ No. of Tetrahedral voids =$\dfrac{1}{6} \times 8 = \dfrac{4}{3}$
Effective number of cation Y=$\dfrac{1}{3} \times $ Octahedral voids =$\dfrac{1}{3} \times 4 = \dfrac{4}{3}$
Hence, the formula is ${X_{\dfrac{4}{3}}}{Y_{\dfrac{4}{3}}}{O_4}$ which is $XY{O_3}$

Note:
It must be noted that the face centered unit cell is sometimes also referred to as cubic close packing or ccp. Also, the arrangement of atoms and the reflecting formula is not the same for all cubic unit cells. The corner and face centered atoms may be the same but the contribution of voids may vary depending on the close packing.